A two component classical Coulomb system is considered, in which particles of charge +q and + 2q are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. At a special value of the coupling constant the correlation functions are calculated exactly and the asymptotic form of the truncated charge-charge correlation is found to obey Jancovici's sum rule.

The paper presents new demonstrably convergent first-order iterative algorithms for unconstrained discrete-time optimal control problems. The algorithms, which solve the linear-quadratic problem in one iterative step, are particularly suited for solving nonlinear problems with linear constraints via penalty function methods. The proofs of the reduction of cost at each iteration and convergence of the algorithms are provided.

The paper extends earlier work by using the factorisation method to discuss solutions of period four for the difference equation

This equation was suggested by R. M. May as a simple mathematical model for the effect of frequency-dependent selection in genetics. It is shown that for a given value of the parameter, a, the identification of solutions of period four can be reduced to finding real roots for a polynomial equation of degree eight. The appropriate values of xn follow from a quartic equation. By splitting up the problem in this way it becomes relatively straightforward to determine the critical values of a at which the various solutions of period four first appear and to discuss the stability of these solutions. Intervals of stability are tabulated in the paper.

In a recent paper the authors give upper and lower bounds for the motion of the moving boundary for the classical Stefan problem for plane, cylindrical and spherical geometries. On comparison with the exact Neumann solution for the plane geometry and no surface radiation the bounds obtained are seen to be quite adequate for practical purposes except for the lower bound at small Stefan numbers. Here improved lower bounds are obtained which in some measure remove this inadequacy. Time dependent surface conditions are also examined and the new lower bounds obtained for the classical problem are illustrated numerically.

In a paper which appeared in this journal, Manocha and Sharma [6] obtained some results of Carlitz [4], Halim and Salain [5] and generalized a few of them by using fractional derivatives. The present paper is concerned with some erroneous results of this paper [6]. Many more sums of the product of hypergeometric polynomials are also obtained.

The evolution of small amplitude waves on an open two layer fluid is investigated. The spatially periodic surface and interface displacements are represented as Fourier series with time dependent coefficients, for which evolution equations with all significant quadratic interactions included, are derived. Solutions to these equations are found analytically for a small number of harmonics, and numerically for a larger number of harmonics. Two numerical solutions are given to illustrate the evolution properties.

This paper considers a single-product industries, fixed capital model discussed briefly by Sraffa in Chaapter X if his book. The analysis is elementary, being based on the direct calculation of a certain martrix inverse. This generalises the approach adopted for circulating capital models. It also demonstrates that common economic and mathematical frameworks exist for both circulating and fixed capital models.

In this paper, we determine the optimal controlled approximation rates from certain bivariate splines on regular meshes.